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G = D100order 200 = 23·52

Dihedral group

Aliases: D100, C4⋊D25, C251D4, C5.D20, C1001C2, D501C2, C2.4D50, C20.2D5, C10.8D10, C50.3C22, sometimes denoted D200 or Dih100 or Dih200, SmallGroup(200,6)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C50 — D100
 Chief series C1 — C5 — C25 — C50 — D50 — D100
 Lower central C25 — C50 — D100
 Upper central C1 — C2 — C4

Generators and relations for D100
G = < a,b | a100=b2=1, bab=a-1 >

50C2
50C2
25C22
25C22
10D5
10D5
25D4
5D10
5D10
2D25
2D25
5D20

Smallest permutation representation of D100
On 100 points
Generators in S100
```(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100)
(1 100)(2 99)(3 98)(4 97)(5 96)(6 95)(7 94)(8 93)(9 92)(10 91)(11 90)(12 89)(13 88)(14 87)(15 86)(16 85)(17 84)(18 83)(19 82)(20 81)(21 80)(22 79)(23 78)(24 77)(25 76)(26 75)(27 74)(28 73)(29 72)(30 71)(31 70)(32 69)(33 68)(34 67)(35 66)(36 65)(37 64)(38 63)(39 62)(40 61)(41 60)(42 59)(43 58)(44 57)(45 56)(46 55)(47 54)(48 53)(49 52)(50 51)```

`G:=sub<Sym(100)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100), (1,100)(2,99)(3,98)(4,97)(5,96)(6,95)(7,94)(8,93)(9,92)(10,91)(11,90)(12,89)(13,88)(14,87)(15,86)(16,85)(17,84)(18,83)(19,82)(20,81)(21,80)(22,79)(23,78)(24,77)(25,76)(26,75)(27,74)(28,73)(29,72)(30,71)(31,70)(32,69)(33,68)(34,67)(35,66)(36,65)(37,64)(38,63)(39,62)(40,61)(41,60)(42,59)(43,58)(44,57)(45,56)(46,55)(47,54)(48,53)(49,52)(50,51)>;`

`G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100), (1,100)(2,99)(3,98)(4,97)(5,96)(6,95)(7,94)(8,93)(9,92)(10,91)(11,90)(12,89)(13,88)(14,87)(15,86)(16,85)(17,84)(18,83)(19,82)(20,81)(21,80)(22,79)(23,78)(24,77)(25,76)(26,75)(27,74)(28,73)(29,72)(30,71)(31,70)(32,69)(33,68)(34,67)(35,66)(36,65)(37,64)(38,63)(39,62)(40,61)(41,60)(42,59)(43,58)(44,57)(45,56)(46,55)(47,54)(48,53)(49,52)(50,51) );`

`G=PermutationGroup([(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100)], [(1,100),(2,99),(3,98),(4,97),(5,96),(6,95),(7,94),(8,93),(9,92),(10,91),(11,90),(12,89),(13,88),(14,87),(15,86),(16,85),(17,84),(18,83),(19,82),(20,81),(21,80),(22,79),(23,78),(24,77),(25,76),(26,75),(27,74),(28,73),(29,72),(30,71),(31,70),(32,69),(33,68),(34,67),(35,66),(36,65),(37,64),(38,63),(39,62),(40,61),(41,60),(42,59),(43,58),(44,57),(45,56),(46,55),(47,54),(48,53),(49,52),(50,51)])`

D100 is a maximal subgroup of   C200⋊C2  D200  D4⋊D25  Q8⋊D25  D1005C2  D4×D25  Q82D25
D100 is a maximal quotient of   Dic100  C200⋊C2  D200  C4⋊Dic25  D50⋊C4

53 conjugacy classes

 class 1 2A 2B 2C 4 5A 5B 10A 10B 20A 20B 20C 20D 25A ··· 25J 50A ··· 50J 100A ··· 100T order 1 2 2 2 4 5 5 10 10 20 20 20 20 25 ··· 25 50 ··· 50 100 ··· 100 size 1 1 50 50 2 2 2 2 2 2 2 2 2 2 ··· 2 2 ··· 2 2 ··· 2

53 irreducible representations

 dim 1 1 1 2 2 2 2 2 2 2 type + + + + + + + + + + image C1 C2 C2 D4 D5 D10 D20 D25 D50 D100 kernel D100 C100 D50 C25 C20 C10 C5 C4 C2 C1 # reps 1 1 2 1 2 2 4 10 10 20

Matrix representation of D100 in GL2(𝔽101) generated by

 62 41 53 9
,
 3 2 97 98
`G:=sub<GL(2,GF(101))| [62,53,41,9],[3,97,2,98] >;`

D100 in GAP, Magma, Sage, TeX

`D_{100}`
`% in TeX`

`G:=Group("D100");`
`// GroupNames label`

`G:=SmallGroup(200,6);`
`// by ID`

`G=gap.SmallGroup(200,6);`
`# by ID`

`G:=PCGroup([5,-2,-2,-2,-5,-5,61,26,1443,418,4004]);`
`// Polycyclic`

`G:=Group<a,b|a^100=b^2=1,b*a*b=a^-1>;`
`// generators/relations`

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